Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

[1]  B. Øksendal,et al.  A maximum principle for optimal control of stochastic systems with delay, with applications to finance. , 2000 .

[2]  Zhen Wu,et al.  Maximum principle for the stochastic optimal control problem with delay and application , 2010, Autom..

[3]  Salvatore Federico,et al.  A stochastic control problem with delay arising in a pension fund model , 2011, Finance Stochastics.

[4]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[5]  Nils Henrik Risebro,et al.  When Are HJB-Equations in Stochastic Control of Delay Systems Finite Dimensional? , 2003 .

[6]  S. Hamadène,et al.  BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations , 2003 .

[7]  Carlo Marinelli,et al.  Stochastic optimal control of delay equations arising in advertising models , 2004, math/0412403.

[8]  Sergei Savin,et al.  On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects , 2007, J. Optimization Theory and Applications.

[9]  Vladimir Borisovich Kolmanovskiĭ,et al.  Control of Systems with Aftereffect , 1996 .

[10]  Shige Peng,et al.  Anticipated backward stochastic differential equations , 2007, 0705.1822.

[11]  B. Øksendal,et al.  Applied Stochastic Control of Jump Diffusions , 2004, Universitext.

[12]  N. Risebro,et al.  WHEN ARE HJB-EQUATIONS FOR CONTROL PROBLEMS WITH STOCHASTIC DELAY EQUATIONS FINITE DIMENSIONAL? , 2001 .

[13]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[14]  Bernt Øksendal,et al.  Optimal control with partial information for stochastic Volterra equations , 2010 .

[15]  Bernt Øksendal,et al.  Some Solvable Stochastic Control Problems With Delay , 2000 .

[16]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[17]  M. Riedle,et al.  Delay differential equations driven by Lévy processes: Stationarity and Feller properties , 2005 .

[18]  B. Larssen.  Dynamic programming in stochastic control of systems with delay , 2002 .

[19]  Situ Rong.  On solutions of backward stochastic differential equations with jumps and applications , 1997 .